S-n-ideals of commutative rings

Let R be a commutative ring with identity and S a multiplicatively closed subset of R. This paper aims to introduce the concept of $S-n$-ideals as a generalization of n-ideals. An ideal I of R disjoint with S is called an $S-n$- ideal if there exists s∈ S such that whenever ab ∈ I for a,~b∈ R, then sa∈ √0 or sb∈ I. The relationships among $S-n$-ideals, n-ideals, S-prime and S-primary ideals are clarified. Besides several properties, characterizations and examples of this concept, S-n-ideals under various contexts of constructions including direct products, localizations and homomorphic images are given. For some particular S and m∈ N, all $S-n$-ideals of the ring Zₘ are completely determined. Furthermore, $S-n$-ideals of the idealization ring and amalgamated algebra are investigated.